Clearly it isn't always possible. For example $5 \times A_4$ has a composition series with factors $2$, $5$, $2$, $3$, in that order. That can't be coarsened to a chief series since $2^2$ is a chief factor.
Are there conditions on either the group or the composition series itself that would guarantee this can be done? Obviously, some strong but uninteresting conditions work, like square-free order, but what about more interesting conditions? For example, is it possible that this can always be done for any composition series in indecomposable finite groups?
Are there any interesting sufficient conditions for infinite groups to have this property?
It doesn't work in general for indecomposable groups. Your example $5 \times A_4$ has faithful irreducible representations of degree $3$ over ${\mathbb C}$, and these can be written for example over a field of order $11$.
Then you can form the semidirect product of the module by the group to get an indecomposable group without the property you are looking for. So, for example, there are such group with the structure $11^3:(5 \times A_4)$, although there might be smaller examples.